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where we have used the fact that
ξ
j
dQ
=−
d
1
for all
j
. The constraint (
1.12
) applied to (
1.13
) is the mathematical rendition of the
desirability of having the average value as the most probable value of the measured
variable.
We now solve (
1.13
) subject to the constraint
N
1
ξ
j
=
0
(1.14)
j
=
by assuming that the
j
th derivative of the logarithm of the probability density can be
expanded as a polynomial in the random error
ln
P
ξ
j
∂ξ
j
∞
∂
k
=
C
k
ξ
j
,
(1.15)
k
=
0
where the set of constants
{
C
k
}
is determined by the equation of constraint
N
∞
k
j
−
C
k
ξ
=
0
.
(1.16)
j
=
1
k
=
0
All the coefficients in (
1.16
) vanish except
k
1, since by definition the fluctuations
satisfy the constraint equation (
1.14
) so the coefficient
C
1
=
=
0 satisfies the constraint.
Thus, we obtain the equation for the probability density
∂
ln
P
(ξ
j
)
∂ξ
j
=
C
1
ξ
j
,
(1.17)
which integrates to
exp
C
1
2
j
P
(ξ
j
)
∝
2
ξ
.
(1.18)
The first thin
g
to notice about this solution is that its extreme value occurs at
ξ
j
=
0, that
=
is, at
Q
j
Q
as required. For this to be a maximum as Gauss required and Simpson
speculated, the constant must be negative,
C
1
<
0, so that the second derivative of
P
at
the extremum is positive. With a negative constant the function decreases symmetrically
to zero on either side, allowing the function to be normalized,
∞
P
(ξ
j
)
d
ξ
j
=
1
,
(1.19)
−∞
and because of this normalization the function can be interpreted as a probability
density. Moreover, we can calculate the variance to be
∞
2
j
2
σ
=
ξ
j
P
(ξ
j
)
d
ξ
j
,
(1.20)
−∞
allowing us to express the normalized probability density as
